Unformatted text preview: pendent. Proof. (3) ( ). Fix A in D. Set ( ) 10 ∫ . Then Theorem Let
be a vector field in an open
connected region D in the xyplane.
(1) If F is conservative then
(2) If D has no holes and , then F is conservative. 11 Example (1) Let
(
) ( ) (a) Check that is conservative. (b) Find . such that
( (c) Evaluate ∫( )
) . 12 Example (2) Evaluate ∫ where C is the curve Remark: Read about “conservation of energy”, p. 10611062. 13 16.4 Green’s theorem
Theorem Suppose C is a simple closed curve in the plane with
positive (counterclockwise) orientation. Assume D is the region
bounded by C. Suppose P and Q have continuous 1st order
partial derivatives on D. Then
∫ ∬( Proof. 14 ) Example (1) Use Green’s theorem to evaluate
∫( (2) Area of D ∬ ) ∫ 15 (3) Find the area bounded by the ellipse
() () (4) Green’s theorem implies that if D has no holes and
on D then
is conservative. 16 16.6 Curl and Divergence
Let 〈 〉 be a vector field in space. Let
〈 (So 〈 〉 〉.) Definition:
curl F  
〈 div F = Example Let 〉
(the divergence of F) 〈 〉. Find div F and curl F. 17 Theorem (1) curl ( ) . (2) If a vector field F is defined on the whole space
and curl F = 0 then F is conservative.
(3) div (curl F) = 0.
Proof. Only (2) is hard to prove. Examples
(1) Let F = 〈
conservative. 〉. Note that curl F 18 . So F is not (2) Let F = 〈
conservative. Then find 〉. Show that F is
such that 19...
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This note was uploaded on 01/31/2014 for the course EAS 207 taught by Professor Richards during the Fall '08 term at SUNY Buffalo.
 Fall '08
 Richards

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